It has to do with the fact that the characteristic classes (over the reals) of a principal $G$-bundle have even degree. We can associate Chern-Simons-like theory to each characteristic class of degree $2k$ together with a $G$-bundle $P$ over a manifold of dimension $2k-1$.
To be a bit more technical a Chern-Simons-like form is asssociated to the following data
1. A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra of $G$ invariant under the action of $G$ by conjugation.
2. A principal $G$-bundle $P\to M$ over $M$.
3. A pair of connections $\nabla^0, \nabla^1$ on $P\to M$.
The Chern-Weil theory produces two closed forms
$$ \Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M) $$
and a form
$$ T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M), $$
such that
$$ d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)-\Phi(\nabla^0). $$
(For details see Chapter 8 of these notes.)
The transgression form $T\Phi(\nabla^1,\nabla^0)$ is the one used in Chern-Simons theories. It depends on two connections, but usually $\nabla^0$ is some fixed connection.
